Well posedness of nonlinear parabolic systems beyond duality

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system $\partial_tu-\mathrm{div} ( ν(|\nabla u|) \nabla u )= -\mathrm{div} f$ with a given {strictly} positive bounded function $ν$, {such that $\lim_{k\to \infty} ν(k)=ν_\infty$} and $f \in L^q$ with $q\in (1,\infty)$. The {existence, uniqueness and regularity} results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty)$.

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